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Paul's critical comment at the end of chapter 2, "We are not like many, peddling the word of God," is our first indication in this letter that there may have been false teachers who had gained some influence in Corinth. Paul's comment could have been considered prideful by his readers, so he quickly reminded them that he needed no self-commendation (as the false teachers likely did). Moreover, Paul and his band needed no letters of reference to gain the Corinthians' trust, and for that matter, needed no letters of reference from the Corinthians to gain the trust of others. The transformed lives of the Corinthians themselves were, metaphorically speaking, Paul's letter of reference, as he was the human instrument whom God used in their transformation. (How many modern pastors would want their flocks to serve as their letter of reference?) Paul made sure his readers knew that he was not boasting in himself, but in the Lord, who made him "an adequate servant of the new covenant" (3:6).
Paul's comparison of the old and new covenants gives us some idea what the false teachers were promoting in Corinth, and it should come as no surprise to us at this point in our study. It was the same old issue of Jewish teachers trying to put Gentile believers under the Law of Moses. Paul was quite bold, saying that the difference between the old and new covenants is death versus life, the letter that kills versus the Spirit that gives life. The only thing that the Law of Moses has ever done for anyone was curse them, since it promised a curse upon transgressors, and no one ever kept it. Thinking one was saved by circumcision, one small requirement of the Mosaic Law, while ignoring the majority of the rest of the Law, was especially ludicrous.
Certainly the Jewish teachers could expound on the glories of how God gave the Law to Moses---recounting how it was written by His finger in stone tablets---as they attempted to convince their audiences of uncircumcised men to line up for some very painful minor surgery. Remember that when Moses carried those stone tablets down from Mt. Sinai, he did not realize that his face was literally glowing with God's glory. Upon seeing him, the people of Israel were afraid to come near him. Consequently, Moses covered his face with a veil to hide the glory that shown from his skin (Ex. 34:27-35). You can imagine Jewish teachers wowing young Gentile men with that story.
Perhaps seizing the Jewish teachers' strongest argument, Paul pointed out that the glory on Moses' face faded, symbolic of the temporary nature of the old covenant, whereas the promised new covenant was never-ending. And there was a second analogy. Moses' veil that hid the glory from Israel symbolized their alienation from God and spiritual darkness, as revealed by their continual rebellion and hardness of heart, even until Paul's day. How tragic it was that people who heard the Law read every Sabbath remained spiritually blind, separated from the glory of God, because they sought salvation in what could only condemn them. When any of them turned to Christ, however, that veil was lifted, revealing the glory that had previously been hidden from them.
So, although the old covenant revealed God's glory, the greater glory was largely hidden and waiting to be fully revealed in Christ and the new covenant. Consequently, the glory of the old was of no comparison to the new, and how foolish it would be for any Gentile believer to listen to those still veiled in darkness who were trying to take them backwards from new life to old death!
I've heard Paul's phrase, "Where the Spirit of the Lord is, there is liberty" (3:17), used to encourage people in church services to abandon their reservations and enjoy the freedom of the Spirit, often an invitation to imitate some bizarre charismatic behavior, an incredible contextual misapplication of Paul's words! Paul was not talking about acting like idiots in church services. The indwelling Spirit, given in the new covenant, grants Jews freedom from the Mosaic Law. Where the Spirit is, there is liberty!